This invention relates generally to the analysis of information which is represented by electrical signals. More particularly, this invention applies to geophysical exploration for petroleum and minerals by means of the seismic technique, whereby acoustic energy is imparted to the earth and the resulting seismic wave which propagates through the earth is reflected and/or refracted at the interfaces of different subsurface geological formations; the reflections and/or refractions are detected, converted to electrical signals, and stored; and the stored electrical signals are analytically processed in order to map the subsurface geological structure. Specifically, this invention is directed to a method and apparatus for processing electrical signals so that the frequency response characteristics, for example, the frequency response of the earth to the acoustic stimulus, can be analyzed in order to yield more precise knowledge, in the example, knowledge about the subsurface geological structure based on the seismic information contained in the electrical signals such that further processing and/or displaying of the information will result in a seismic section of improved resolution.
As is well known, any electrical signal, no matter how complex, is comprised of a number of fundamental component signals of different amplitudes and frequencies which, when combined, form the electrical signal. Today many electrical signals, including most seismic information obtained in the field, is in digital form. Digital filtering is a process using digital techniques whereby the component signals within a selected range of frequencies can be selected, that is, sorted, from the larger suite of component signals of the electrical signal.
Now, the purpose for digital filtering can be addressed. Generally, the electrical signal contains some informational signal of interest plus additive noise, that is, the informational signal is contaminated by noise. The noise contamination impacts the susceptibility of the electrical signal to analysis in order to extract the informational signal.
The use of digital filtering in information processing is now well established in such diverse fields as communications, radar, etc. Digital filtering has also been used in the geophysical field (Kulhanek, Ota, 1976, Introducton to Digital Filtering in Geophysics: Amsterdam, Elsevier Scientific Publishing Co.). Digital filtering in the geophysical field is now very broad and can be divided into more specialized subfields, such as predictive deconvolution (Peacock, K. L., and Treitel, S., 1969, Predictive Deconvolution: Theory and Practice: Geophysics, v. 39, pp. 155-169), adaptive filtering (Widrow, B., McCool, J. M., Larimore, M. G., and Johnson, C. R., 1976, Stationary and Nonstationary Learning Characteristics of the LMS Adaptive Filter: IEEE Proceedings, v. 64, pp. 1151-1162), two-dimensional filtering (Clement, W. G., 1973, Basic Principles of Two-Dimensional Digital Filtering: Geophysical Prospecting, v. 21, pp. 125-145), and time-variant filtering (Lenithan, U.S. Pat. No. 3,701,091). Development and application of digital filtering in the geophysical field have expanded rapidly in recent years as the use of computers has become more widespread and filtering processes have been performed by computer.
Time-variant filtering is a process for sorting the component signals from the electrical signal wherein the filtering process is modified over time. Time-variant filtering is sometimes necessary to analyze electrical signals which themselves have a time-variant character. For example, seismic information in the form of electrical signals often exhibits a progressive loss of high-frequency content over time (see above-mentioned U.S. Pat. No. 3,701,091 to Lenihan and Gurbuz, B. M., 1972, Signal Enhancement of Vibratory Source Data in the Presence of Attenuation: Geophysical Prospecting, v. 20, pp. 421-438, at page 428).
Considered in greater detail, during the course of seismic prospecting, such as the mapping of the subsurface geological structure by creating a seismic wave and observing the arrival times of the wave reflected from acoustic-impedance contrasts or refracted through high-velocity members, the amplitude of the wave decays as it propagates through the earth. Furthermore, the amplitude decay will be frequency dependent in that the higher frequency seismic wave components tend to suffer greater amplitude attenuation, particularly at later arrival times. Generally, several factors are viewed as contributing to the amplitude attenuation, such as geometrical spreading, reflection absorption, scattering, and various other acoustic transmission loss mechanisms.
A technique for recognizing amplitude decay as a function of time and frequency is to perform a Fourier analysis of the seismic information and view the phenomenon in the frequency domain. In particular, it is observed that the frequency content along the length of the seismic section will shift to lower frequencies as time increases. This evidences that higher frequency seismic wave components are attenuated at a faster rate than lower frequency components. Therefore, the earlier acquired seismic wave reflections and/or refractions, which represent the shallow subsurface geological formations, have a higher mean frequency than the later acquired reflections and/or refractions, which represent the deep subsurface formations. Frequency dependent absorption with depth (that is, time of arrival) not only creates a problem in interpreting observed seismic wave reflections and/or refractions, but also has an analogous influence when it is attempted to construct a priori a synthetic seismic section. For a theoretical discussion of the effect of frequency in depth dependent absorption when constructing a synthetic seismic section see Trovey, A. W., 1962, Theoretical Seismograms with Frequency and Depth Dependent Absorption: Geophysics, v. 27, pp. 766-785.
Now, recorded seismic information always includes some background (ambient) noise in addition to the detected seismic waves reflected and/or refracted from the subsurface geological formation (referred to as the "seismic signal"). Noise originates from many sources, such as atmospheric electromagnetic disturbances, wind, motor vehicle traffic in the vicinity of the prospect area, recorder electrical noise, etc. The noise contaminates the seismic signal. Unfortunately, the amplitude of the noise on each seismic record is not known. Only the amplitude of the seismic signal plus the noise is known.
Digital filtering has traditionally been applied for rejecting frequency components outside the band of interest in order to reduce noise, that is, improve the signal-to-noise ratio. In accordance with the present invention, filtering is caused to reject increasingly lower frequencies with increasing time in recognition of the phenomenon that higher frequency components of the seismic signal are increasingly attenuated over time, and therefore, the seismic signal components at higher frequencies are virtually nonexistent in the later portion of the seismic record; that is, higher frequencies present in the later portion of the seismic record are dominated by noise.
Lenihan, U.S. Pat. No. 3,701,091, mentioned above, discloses a digital technique for time-varying filtering of seismic signals which preferably involves three steps. The initial step is selecting a set of individual digital filters having predetermined start times. This set of digital filters serves as the input. The start times, as well as the passband, of these input filters are determined by analyzing the seismic signals, for example, by frequency analysis, in order to ascertain the frequency of the signals and where the frequency changes occur, whereby it can be determined where changes are needed in the filters to adapt to the frequency changes. Next, a set of individual digital time-variant filters is formed from the input set of filters; that is, the center frequency and envelope frequency of the input filters are determined, and additional individual digital filters are interpolated between adjacent input filters based on the difference in center frequency between adjacent input filters. Then, the time responses of the filters are determined and truncated.
Unfortunately, the technique disclosed by Lenihan, U.S. Pat. No. 3,701,091, requires that a high number of interpolated filters be interposed between the input filters in order to obtain a smoothly varying set of filters over time, as otherwise there will be discontinuities in the filtered seismic signals. Such an approach necessitates a lengthy computation time, since each of the interpolated filters is calculated. Furthermore, the calculations themselves are complex, and therefore, even in a case where a relatively low number of filters is interpolated and the smoothness of the filtering is compromised, computation time is still considerable. The present invention provides a method and apparatus for digital time-varying filtering which obtains smooth, continuous filtered seismic signals and at the same time is less complex than disclosed by Lenihan, U.S. Pat. No. 3,701,091, and requires less computation time.
Cain, G. D., 1972, Hilbert Transform Description of Linear Filtering: Electronics Letters, v. 8, pp. 380-382, discloses analog time-invariant Hilbert transform filters. Roy, S. C. D., and Agarwal, A., 1978, Digital Low-Pass Filtering Using the Discrete Hilbert Transform: IEEE Trans., v. ASSP-26, pp. 465-467, and Sabri, M. S., and Steenaart, W., 1977, Discrete Hilbert Transform Filtering: IEEE Trans., v. ASSP-25, pp. 452-454, describe digital Hilbert transform filters. However, these digital Hilbert transform filters are not time-variant. The above-mentioned article by Cain suggests time-variant Hilbert transform filters. One author has discussed time-variant Hilbert transform filters, Saraga, W., 1967, New Class of Time-Varying Filters: Electronic Letters, v. 3, pp. 158-160. These time-variant Hilbert transform filters are analog filters rather than digital. The present invention deals with digital filters based on the Hilbert transform that permit time-variant filtering to be easily implemented. Because of the physical limitations associated with analog filters, the digital filters of the invention are more flexible with respect to minimizing phase distortion and with respect to achieving high accuracy.